How should I concretely think of the etale cohomology of $\text{Spec }k$, where $k$ a field? Is it possible to get some concrete examples? This is apparently connected (a reformulation in another language) to Galois theory, and we have concrete examples for that, so it would be helpful to me if I could make some analogies. I am new to etale cohomology, but I will outline what I know below. Please tell me if there are any mistakes in my understanding.

So we would like to put a sheaf (of abelian groups) on $\text{Spec }k$. But this is etale cohomology, and instead of open sets, we have instead etale $k$-algebras, which are products of finitely many separable field extensions of $k$. So instead of inclusions of open sets, we have etale morphisms

$k\rightarrow k_{i}$

where $k_{i}$ is an etale $k$-algebra which lead to the map of 1-point (since the spectrum of a field is composed of only the generic point) topological spaces

$\text{Spec }k_{i}\rightarrow \text{Spec }k$

and restriction maps

$\mathcal{F(k)}\rightarrow \mathcal{F(k_{i})}$.

Intuitively, what are the abelian groups $\mathcal{F}(k)$ and $\mathcal{F}(k_{i})$? Under what law of composition? Do they have anything to do with the generators of the field extension, or the basis we use when we express the field extension as a vector space? Any concrete examples?

The sheaf $\mathcal{F}(k_{sep})$, where $k_{sep}$ is a separable closure of $k$, is then defined as the direct limit (the stalk)

$\mathcal{F}(k_{sep})$=$\varinjlim \mathcal{F}(k_{i})$

Why must we define $\mathcal{F}(k_{sep})$ as a direct limit? How do the $k_{i}$ form a directed system? What are the morphisms I must use to define the direct limit?

Finally, we have

$\mathcal{F}(k)$=$\mathcal{F}(k_{i})^{\text{Gal }(k_{i}/k)}$ i.e. ($\mathcal{F}(k)$ is the subgroup of $\mathcal{F}(k_{i})$ that is fixed under the action of $\text{Gal }(k_{i}/k))$.

How is this so? It seems I would need the answers to the previous questions first before I can get this one. I am guessing this will be generalized to become a reformulation of the main theorem of Galois theory.


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